Decay of CP-even Higgs $H\rightarrow h \gamma \gamma$ in Two Higgs Doublet Model: (I) one-loop analytic results, ward identity checks

We present the first analytical expressions for one-loop induced contributions for the decay channels of CP-even Higgs $H\rightarrow h \gamma \gamma$ with $h$ being standard model-like Higgs boson within the framework of Two Higgs Doublet Model in this paper. One-loop form factors for the decay processes are written in terms of the scalar Passarino-Veltman functions following the general notations of the package~{\tt LoopTools} as well as the library {\tt Collier}. Subsequently, physical results for the decay processes can be generated numerically by using one of the above-mentioned packages. The analytical expressions shown in this paper, are verified by several numerical checks, for examples, the ultraviolet (UV) and the infrared (IR) finiteness for one-loop amplitude. Furthermore, the amplitude must be followed the so-called ward identity due to on-shell photons in final states. The identity can also be tested numerically in this work. We find that the numerical results for the checks are good stability. In phenomenological studies, the differential decay rates as functions of the invariant of two photons in final state of $H\rightarrow h \gamma \gamma$ are first studied in parameter space for all types of Two Higgs Doublet Models.